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Accretion of dark matter by stars

Richard Brito1, Vitor Cardoso1,2 Hirotada Okawa3,41 CENTRA, Departamento de Fısica, Instituto Superior Tecnico – IST,

Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal2 Perimeter Institute for Theoretical Physics Waterloo, Ontario N2J 2W9, Canada

3 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan and4 Advanced Research Institute for Science & Engineering,

Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan

Searches for dark matter imprints are one of the most active areas of current research. We focushere on light fields with mass mB, such as axions and axion-like candidates. Using perturbativetechniques and full-blown nonlinear Numerical Relativity methods, we show that (i) dark mattercan pile up in the center of stars, leading to configurations and geometries oscillating with frequencywhich is a multiple of f = 2.5 × 1014

(

mBc2/eV

)

Hz. These configurations are stable throughoutmost of the parameter space, and arise out of credible mechanisms for dark-matter capture. Starswith bosonic cores may also develop in other theories with effective mass couplings, such as (massless)scalar-tensor theories. We also show that (ii) collapse of the host star to a black hole is avoided byefficient gravitational cooling mechanisms.

PACS numbers: 95.35.+d,04.40.-b,12.60.-i,04.25.D-

I. Introduction. The standard picture for the evolu-tion and structure of our universe relies on the existenceof unseen forms of matter, generically called dark mat-ter (DM). The evidence for DM in observations is over-whelming, starting with galaxy rotation curves, gravi-tational lensing and the cosmic microwave background.While carefully concocted modified theories of gravitycan perhaps explain almost all observations, the most at-tractive and accepted explanation lies in DM being com-posed mostly of cold, collisionless particles.Ultralight fields, such as axions or axion-like candi-

dates are an attractive possibility [1–3]. Axions wereoriginally devised to solve the strong-CP problem, but re-cently a plethora of other, even lighter fields with masses10−10−10−33 eV/c2, have also become an interesting pos-sibility, in what is commonly known as the axiverse sce-nario [4]. The simplest possible theory is that of a mas-sive scalar φ or vector Aµ minimally coupled to gravity,and described by the Lagrangian

L =R

κ− F 2

4− µ2

V

2AνA

ν − gµν

2φ∗

,µφ,ν − µ2S

2φ∗φ . (1)

We take κ = 16π, Fµν ≡ ∇µAν − ∇νAµ is the Maxwelltensor and F ≡ FµνFµν . The mass mB of the boson un-der consideration is related to the mass parameter abovethrough µS,V = mB/~, and the theory is controlled bythe dimensionless coupling

G

c~MTµS,V = 7.5 · 109

(

MT

M⊙

)(

mBc2

eV

)

, (2)

where MT is the total mass of the bosonic configuration.It is appropriate to emphasize that DM has not been

seen nor detected through any of the known standardmodel interactions. The only evidence for DM is throughits gravitational effect. Not surprisingly, the quest forDM is one of the most active fields of research of this cen-tury. Because DM interacts feebly with Standard Model

particles, and thanks to the equivalence principle, themost promising channel to look for DM imprints consistsof gravitational interactions. It turns out that classical

0 10 20 30 40 50µR

0

0.2

0.4

0.6

0.8

1.0

1.2µM

T

vector oscillatonscalar oscillatonscalar boson star

FIG. 1. Comparison between the total mass of a boson star(complex scalar) and an oscillaton (real scalar or vector fields),as a function of their radius R. R is defined as the radiuscontaining 98% of the total mass. The procedure to find thediagram is outlined in the main text.

massive bosonic fields minimally coupled to gravity, de-scribed by Eq. (1) – which will be our working model forDM here – can form structures [5–9]. Self-gravitatingcomplex scalars may give rise to static, spherically-symmetric geometries called boson stars, while the fielditself oscillates [5, 6] (for reviews, see Refs. [10–13]).On the other hand, real scalars have a non-trivial time-dependent stress-tensor but may give rise to long-termstable oscillating geometries called oscillatons [7]. Bothsolutions arise naturally as the end-state of gravitationalcollapse [7, 14, 15], and both structures share a similarmass-radius relation, summarized in Fig. 1. In this figurewe also show the mass-radius relation for massive vectors,which as far as we know has not been discussed before.

2

Boson stars and oscillatons have a maximum massMmax,given approximately by

Mmax

M⊙

= 8× 10−11

(

eV

mBc2

)

, (3)

for scalars and slightly larger for vectors. Oscillatonsare not truly periodic solutions of the field equations,as they decay through quantum and classical processes.However, their lifetime Tdecay is extremely large for themasses of interest [16, 17],

Tdecay ∼ 10324(

1meV

mBc2

)11

yr . (4)

II. Compact stars with DM cores. Although notframed in the context of DM capture, some stars withcomplex scalar fields in their interior were considered be-fore in the literature [18–25]. Our construction is moregeneric.Consider the Lagrangian (1), augmented with the

stress-energy tensor for a perfect fluid,

T µνfluid = (ρF + P )uµuν + Pgµν , (5)

with uµ the fluid four-velocity, uµ =(

√

−1/gtt, 0, 0, 0)

,

and ρF , P its density and pressure. It is easy toshow that the vector field must satisfy the constraintµ2V ∇µA

µ = 0, while the Bianchi identities and theKlein-Gordon equation, impose the conservation equa-tions ∇µT

µνfluid = 0. Let us focus for simplicity on scalar

fields. For complex scalars, self-gravitating solutionswere considered many times, consisting of sphericallysymmetric, static boson stars [10–13]. In Ref. [7] it wasshown that real massive scalars admit oscillating self-gravitating solutions. Consider therefore a general time-dependent, spherically symmetric geometry

ds2 = −F (t, r)dt2 +B(t, r)dr2 + r2dΩ2 . (6)

Rescaling the scalar field as φ → φ/√8π and defining

the function C(t, r) = B(t, r)/F (t, r), the field equationsthen lead to the system of partial differential equations

B′/B = (r/2)[

Cφ2 + (φ′)2 +B(

µ2Sφ

2 + 16πρF)

]

+(1−B)/r , (7)

C′/C = 2/r +Br(

µ2Sφ

2 + 8πρF − 8πP)

− 2B/r , (8)

φC = −Cφ/2 + φ′′ + 2φ′/r − C′φ′/(2C)−Bµ2Sφ , (9)

2P ′ = − (P + ρF ) (CB′ −BC′) /(BC) . (10)

These equations suggest the following periodic expansion

N i =

∞∑

j=0

N i2j(r) cos (2jωt) ,

φ(t, r) =

∞∑

j=0

φ2j+1(r) cos [(2j + 1)ωt] , (11)

where N i = (B(t, r), C(t, r), ρF (t, r), P (t, r)). The equa-tions of motion need to be supplemented by an equationof state. We will focus exclusively, and for simplicity, ona polytropic equation of state P = KρΓF with K = 100and Γ = 2, which can mimic neutron stars [26], but ourresults generalize to other equations of state 1. Insertingthe expansion (11) into the system (7)– (10) and trun-cating the series at a given j 2, gives a set of ordinarydifferential equations for the radial Fourier componentsof the metric functions and the scalar field. These equa-tions need to be solved imposing regularity at the originand asymptotic flatness.A useful quantity to describe scalar-fluid stars is the

scalar field’s energy density, given by

2ρφ = −2T 00 = φ2/gtt + φ

′2/grr + µ2Sφ

2 . (12)

With this, we define the time-average total mass in thefluid and bosons as

MF,B =

∫ ∞

0

4π⟨√

B ρF,B

⟩

r2dr , (13)

where <> denotes a temporal average. The total massMT can be found in the usual way through the met-ric component grr which asymptotically approaches theSchwarzschild solution at infinity. The procedure to findfluid-vector stars is similar and will not be described fur-ther. Figure 1 shows a phase diagram of solutions whenthere is no fluid. We find that the maximum mass fora stable scalar oscillaton is MT ∼ 0.6/µS in agreementwith previous studies [7, 28]. As far as we know, thevector solution has not been described anywhere else.When the fluid is present, the solutions depend on two

parameters ρF 0(0) and φ1(0), and can be characterizedby the mass coupling µSM0, where M0 is the mass ofthe static star for vanishing scalar field, corresponding tothe same value of central density ρF 0(0). Different con-figurations are shown in Fig. 2, where we compare thescalar energy density against the fluid’s density. In allcases we also show the density profile when the scalarfield is trivial, corresponding to the same fluid mass MF .The overall behavior might have been anticipated froman analysis of Fig. 1: for light fields, µSM0 < 1, thescalar profile is extended and the pure oscillaton solutionis broad and light. As such, the scalar has a negligible in-fluence on the fluid distribution (as can be seen from thefact that the zero-scalar line ρφ=0 overlaps with the fluidline), and these stars simply have an extended scalar con-densate protruding away from them. In fact, our resultsare compatible with a decoupling between the boson and

1 In geometrical units G = c = 1 and Γ = 2, K has units length2

and can be used to set the length-scale of the problem in theabsence of the scalar field [26]. The choice K = 100 was used ine.g. Ref. [27].

2 We find that the solutions typically converge already for j = 2for most of the parameter space.

3

0 5µSr

0

1

2

3

4

5

6

7

8

9

10

ρF (x104)

ρφ(x107)

ρφ=0(x104)

0 5 10 15µSr

ρF (x104)

ρφ (x104)

ρφ=0 (x104)

0 50 100 150 200µSr

ρF (x104)

ρφ (x103)

ρφ=0 (x104)

FIG. 2. Comparison between the (time average) energydensity of the scalar field ρφ and the fluid ρF for mixedscalar oscillatons and fermion fluids. From left to right,µSM0 = 0.1, MB/MT ≈ 21%, µSM0 = 1, MB/MT ≈ 5%,andµSM0 = 10, MB/MT ≈ 5%. Squares denote the correspond-ing quantities for complex fields (i.e. mixed boson-fluid stars).The overlap is nearly complete. Here M0 and ρφ=0 are thetotal mass of the star and the energy density of the fluid, re-spectively, when the scalar field vanishes everywhere. In theleft panel, the ρφ=0 and the ρF lines are indistinguishable,because light fields have a negligible influence on the fluiddistribution.

0 20 40 60 80 100 120 140µSr

-3

-2

-1

0

1

2

3

4

5

6

ρF 0 (x104)

ρF 2 (x104)

ρF 4 (x106)

FIG. 3. Density profile of the fluid at t = 0 and its firstFourier components for µSM0 = 10 and MB/MT ≈ 5%.

fluid for large µSMB. For this case Fig. 1 alone is enoughto interpret the bosonic distribution. For example, forµSM0 = 10,MB/MT = 5%, we get µSMB ∼ 0.3, whichwould imply from Fig. 1 that µSR ∼ 20 for the scalarfield distribution. This is indeed apparent from Fig. 2.Similar conclusions were reached when studying mixedfermion fluid/boson stars with complex fields [20] 3.Global thermalization. We should stress that the com-posite stars we study here in general oscillate, drivenby the scalar field. For these configurations, the con-vergence of the series (11) is rather fast, meaning that

3 In fact, the structure of mixed oscillatons and fluid stars is almostidentical to that of geons and fluids, as can be seen from Fig. 2,where we overplot with dotted lines the complex field case.

the amplitude of the oscillations is relatively small, butnonzero. For high scalar field central densities, first-orderterms j = 1 in the density might become of the order ofthe zeroth-order term, making it difficult to find theseconfigurations with good accuracy. For a given ρF 0(0),we expect these configurations to become unstable atsome threshold φ1(0) > φc

1(0). Our results are consistentwith what was previously found for boson-fermion fluidstars [18, 19]. Although field configurations with highMB/MT are more challenging to find for large µSM0,we expect them to follow the same kind of behavior asthat found in boson-fermion stars. An example showingthe amplitude of the oscillations is shown in Fig. 3 forM0µS = 10, for which the star is oscillating with largeamplitudes (the oscillating component is of the same or-der of magnitude as the static one). We find that evenfor MB/MT = 0.01, and for µSM0 = 10 the oscillationsare of order 10% of the static component. These oscilla-tions imply that both the scalar field and the fluid density(which is coupled to it gravitationally), vary periodicallywith a frequency 4

f = 2.5× 1014(

mBc2

eV

)

Hz , (14)

or multiples thereof. For axion-like particles with masses∼ 10−5 eV/c2, these stars would emit in the microwaveband. These oscillations are driven by the boson coreand might have observable consequences; it is in principleeven possible that resonances occur when the frequencyof the scalar is equal to the oscillation frequency of theunperturbed star. The joint oscillation of the fluid andthe boson might be called a global thermalization of the

star, and is expected to occur also for boson-star-likecores (which give rise to static boson cores), once thescalar is allowed to have non-zero couplings with the starmaterial.We have neglected viscosity in the star’s fluid and local

thermalization. Viscous timescales for neutron star os-cillations can be shown to be large compared to the stardynamical timescale R, but small when compared to the(inverse of) the accretion rate likely to be found in anyrealistic configuration [29]. As such, we expect that vis-cosity will damp global oscillations of the star, eventuallyleading to a depletion of the scalar field core. A similareffect will occur with local thermalization of the scalarwith the star material; detailed studies of these effectsare still necessary.Stability. Finally, the stability of these solutions cannoteasily be inferred from a mass versus radius diagram, andit requires a dynamical analysis which goes beyond thescope of this Letter. However, one can argue that a nec-

essary condition for stability is that the binding energyMT −MF −MB be negative [19]. Although negative val-ues do not necessarily imply stability, they do give strong

4 The fundamental frequency ω ∼ µS, V [7, 16].

4

time: t/w=12.0

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x/w

0

0.5

1

1.5

2

2.5

3

z/w

10-4

10-3

10-2

10-1

100

ρw2

time: t/w=36.0

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x/w

0

0.5

1

1.5

2

2.5

3

z/w

10-4

10-3

10-2

10-1

100

ρw2

time: t/w=44.0

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x/w

0

0.5

1

1.5

2

2.5

3

z/w

10-4

10-3

10-2

10-1

100

ρw2

FIG. 4. Snapshot of density ρ during the collision of two equal-mass oscillatons, each with MµS ∼ 0.5, RµS ∼ 15, along theplane of collision, for half the space (the remaining space can be obtained by symmetry). Notice how the final configuration ismore compact, and massive. At very late times, we find that it relaxes to a point on the curve of Fig. 1.

support to the claim that these configurations are sta-ble. A more careful stability analysis shows that for suf-ficiently small φ1(0) and for stars which are stable in theabsence of scalars, the negativity of the binding energyis a good criterion for stability [30]. To summarize, forsufficiently small scalars composite stars are dynamicallystable. On the other hand, our results show that theseconfigurations can be understood well from the mass-radius relation of oscillatons. The maximum mass sup-ported is (3), Mmax/M⊙ = 8 × 10−11 eV/(mBc

2), whichfor a neutron star and an axion field of mass 10−5 eVfalls well within the stability regime [31].III. Formation: DM accretion, core growth and

gravitational cooling mechanism. We have shownthat pulsating stars with DM cores exist as solutions ofthe field equations. Do they form dynamically? Pulsat-ing purely bosonic states certainly do [7, 14, 15]. Thereare two different channels for formation of compositefluid/boson stars. One is through gravitational collapsein a bosonic environment, through which the star is bornalready with a DM core. The second process consists ofcapture and accretion of DM into the core of compactstars [32, 33].Once the scalar is captured it will interact with the bo-

son core. Interactions between complex fields have shownthat equal mass collisions at low energies form a boundconfiguration [34, 35]. In other words, two bosonic corescomposed of complex fields interact and form a moremassive core at the center. This new bound configu-ration is in general asymmetric and will decay on largetimescales [13], the final state being spherical [36]. Wehave repeated this analysis for two oscillatons collidingat sufficiently small energies, using the methods of Nu-merical Relativity outlined in Refs. [15, 37]. We findthat generically the collision of two oscillatons whose to-tal mass is below the peak value will give rise to a morecompact, more massive oscillaton configuration. By con-trast, the collision of oscillatons whose mass exceeds thepeak value results in large mass losses and a lighter finalconfiguration. An example of such a collision is shown inFig. 4 for equal-mass oscillatons each with MTµS ∼ 0.5(close to the peak value of Fig. 1). Note that the totalmass is larger than the peak value and one would naively

predict gravitational collapse to a black hole (BH). In-stead, the collision first produces a massive, more com-pact oscillating object, accompanied by large scalar fieldlosses, as is apparent from Fig. 4. On large timescalesthe configuration relaxes to a very low-mass structure.This is a general feature of what has been termed

the “gravitational cooling mechanism:” a very efficient(dissipationless) mechanism that stops them from grow-ing past the unstable point, through the ejection ofmass [28, 38]. Such features have been observed in thepast in other setups, such as spherically symmetric grav-itational collapse (see Fig. 2 in Ref. [15]), or slightly per-turbed oscillatons [28]. Gravitational cooling provides acounter-example to an often-used assumption in the liter-ature: that stars accreting DM will grow past the Chan-drasekhar limit for the DM core and will collapse to aBH [33, 39–42]. Our results show that this need not bethe case, if the DM core is prevented from growing by aself-regulatory mechanism, such as gravitational cooling.Thus, even though other more detailed simulations are

still needed, the likely scenario for evolution would com-prise a core growth through minor mergers, slowing downclose to the mass-radius peak (see Fig. 1), at which pointit stops absorbing any extra bosons [15, 28]. In otherwords, the unstable branch is never reached. This phe-nomenology is specially interesting, as it would also pro-vide a capture mechanism for these fields which is inde-pendent of any putative nucleon-axion interaction crosssection: as we discussed, the bosonic core grows (in mass)through accretion until its peak value. At its maximum,it has a size RB/M⊙ ∼ MB/M⊙. This is the bosoniccore minimum size, as described by Fig. 1. Even forMB = 0.01MT the core has a non-negligible size and isable to capture and trap other low-energy oscillatons.IV. Conclusions. Previous works on the subject of DMaccretion by stars have implicitly assumed that the DMcore is able to grow without bound and eventually col-lapse to a BH [33, 39–42]. Our results, from full nonlinearsimulations of the field equations, show that the core maystop growing when it reaches a peak value, at the thresh-old of stability, if DM is composed of light massive fields.Gravitational cooling quenches the core growth for mas-sive cores and the core growth halts, close to the peak

5

value (c.f. Fig. 1).Finally, our results are formally valid only for the the-

ory (1), and require the fields to be massive. Minimallycoupled, multiple (real) scalars, interacting only gravi-tationally, were also shown to give rise to similar con-figurations [43]. We have explicitly verified that similarresults may hold in other setups. For example, scalar-tensor theories (which can be shown to give rise to aneffective position-dependent mass term [44, 45]) and non-minimally coupled massless scalars are, in principle, alsoable to develop nontrivial configurations with a scalarizedcore. Further details will be presented in a forthcomingwork.Acknowledgments. We thank E. Berti, L. Gualtieri,I. Lopes, D. Marsh and U. Sperhake for useful com-ments and feedback. R.B. acknowledges financial sup-

port from the FCT-IDPASC program through thegrant SFRH/BD/52047/2012, and from the FundacaoCalouste Gulbenkian through the Programa Gulbenkiande Estımulo a Investigacao Cientıfica. V.C. acknowledgesfinancial support provided under the European Union’sFP7 ERC Starting Grant “The dynamics of black holes:testing the limits of Einstein’s theory” grant agreementno. DyBHo–256667, and H2020 ERC Consolidator Grant“Matter and strong-field gravity: New frontiers in Ein-stein’s theory” grant agreement no. MaGRaTh–646597.Research at Perimeter Institute is supported by the Gov-ernment of Canada through Industry Canada and bythe Province of Ontario through the Ministry of Eco-nomic Development & Innovation. We acknowledge allo-cations on SDSC Trestles and Comet and TACC Stam-pede through NSF-XSEDE Grant PHY-090003.

[1] D. J. E. Marsh and A.-R. Pop, Mon.Not.Roy.Astron.Soc.451, 2479 (2015), [1502.03456].

[2] H.-Y. Schive, T. Chiueh and T. Broadhurst, NaturePhysics 10, 496 (2014), [1406.6586].

[3] H.-Y. Schive et al., Physical Review Letters 113, 261302(2014), [1407.7762].

[4] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloperand J. March-Russell, Phys.Rev. D81, 123530 (2010),[0905.4720].

[5] D. J. Kaup, Phys. Rev. 172, 1331 (1968).[6] R. Ruffini and S. Bonazzola, Phys. Rev. 187, 1767

(1969).[7] E. Seidel and W. Suen, Phys.Rev.Lett. 66, 1659 (1991).[8] N. Banik, A. J. Christopherson, P. Sikivie and E. M.

Todarello, 1504.05968, for a discussion on the validity ofthe classical approximation.

[9] A. H. Guth, M. P. Hertzberg and C. Prescod-Weinstein,1412.5930.

[10] P. Jetzer, Phys. Rep. 220, 163 (1992).[11] F. Schunck and E. Mielke, Class. Quant. Grav. 20, R301

(2003), [0801.0307].[12] S. L. Liebling and C. Palenzuela, Living Rev. Rel. 15, 6

(2012), [1202.5809].[13] C. F. Macedo, P. Pani, V. Cardoso and L. C. B. Crispino,

Phys.Rev. D88, 064046 (2013), [1307.4812].[14] D. Garfinkle, R. B. Mann and C. Vuille, Phys.Rev. D68,

064015 (2003), [gr-qc/0305014].[15] H. Okawa, V. Cardoso and P. Pani, Phys.Rev. D90,

104032 (2014), [1409.0533].[16] D. N. Page, Phys.Rev. D70, 023002 (2004), [gr-

qc/0310006].[17] P. Grandclement, G. Fodor and P. Forgacs, Phys.Rev.

D84, 065037 (2011), [1107.2791].[18] A. Henriques, A. R. Liddle and R. Moorhouse, Phys.Lett.

B233, 99 (1989).[19] A. Henriques, A. R. Liddle and R. Moorhouse,

Nucl.Phys. B337, 737 (1990).[20] L. Lopes and A. Henriques, Phys.Lett. B285, 80 (1992).[21] A. B. Henriques and L. E. Mendes, Astrophys.Space Sci.

300, 367 (2005), [astro-ph/0301015].[22] K. Sakamoto and K. Shiraishi, Phys.Rev. D58, 124017

(1998), [gr-qc/9806040].

[23] C. M. de Sousa and V. Silveira, Int.J.Mod.Phys. D10,881 (2001), [gr-qc/0012020].

[24] C. M. de Sousa and J. Tomazelli, Phys.Rev. D58, 123003(1998), [gr-qc/9507043].

[25] F. Pisano and J. Tomazelli, Mod.Phys.Lett. A11, 647(1996), [gr-qc/9509022].

[26] R. F. Tooper, ApJ140, 434 (1964).[27] S. Valdez-Alvarado, C. Palenzuela, D. Alic and

L. A. Urena-Lopez, Phys.Rev. D87, 084040 (2013),[1210.2299].

[28] M. Alcubierre et al., Class.Quant.Grav. 20, 2883 (2003),[gr-qc/0301105].

[29] C. Cutler, L. Lindblom and R. J. Splinter, ApJ363, 603(1990).

[30] A. Henriques, A. R. Liddle and R. Moorhouse, Phys.Lett.B251, 511 (1990).

[31] M. Gleiser, Phys.Rev.D 38, 2376 (1988).[32] W. H. Press and D. N. Spergel, Astrophys.J. 296, 679

(1985).[33] A. Gould, B. T. Draine, R. W. Romani and S. Nussinov,

Phys.Lett. B238, 337 (1990).[34] A. Bernal and F. Siddhartha Guzman, Phys.Rev. D74,

103002 (2006), [astro-ph/0610682].[35] C. Palenzuela, I. Olabarrieta, L. Lehner and S. L.

Liebling, Phys.Rev. D75, 064005 (2007), [gr-qc/0612067].

[36] A. Bernal and F. S. Guzman, Phys.Rev. D74, 063504(2006), [astro-ph/0608523].

[37] H. Okawa, H. Witek and V. Cardoso, Phys.Rev. D89,104032 (2014), [1401.1548].

[38] E. Seidel and W.-M. Suen, Phys.Rev.Lett. 72, 2516(1994), [gr-qc/9309015].

[39] G. Bertone and M. Fairbairn, Phys.Rev. D77, 043515(2008), [0709.1485].

[40] S. D. McDermott, H.-B. Yu and K. M. Zurek, Phys.Rev.D85, 023519 (2012), [1103.5472].

[41] P. W. Graham, S. Rajendran and J. Varela, 1505.04444.[42] C. Kouvaris and P. Tinyakov, Phys.Rev. D90, 043512

(2014), [1312.3764].[43] S. H. Hawley and M. W. Choptuik, Phys.Rev. D67,

024010 (2003), [gr-qc/0208078].[44] V. Cardoso, I. P. Carucci, P. Pani and T. P. Sotiriou,

6

Phys.Rev.Lett. 111, 111101 (2013), [1308.6587].[45] V. Cardoso, S. Chakrabarti, P. Pani, E. Berti and

L. Gualtieri, Phys.Rev.Lett. 107, 241101 (2011),[1109.6021].